Past the Natural Boundary: Three Continuations of a Mock Theta Radial Limit, the Grip on Gamma and p-adic L-Values, and a Formalization Programme

Ramanujan's third-order mock theta function and its parity-shifted companion have the unit circle as a genuine natural boundary. We prove the boundary is nonetheless permeable in three precise senses - spectral (Mellin-Gamma), resurgent (Borel/Lerch transcendent), and p-adic (Coleman polylogarithm) - all governed by one radial moment kernel. We give exact proofs of the inversion duality, the Frobenius identity, and the moment recurrence, and identify the p-adic continuation with Dirichlet p-adic L-values via Coleman's theorem. A companion Lean 4 repository (included) machine-verifies four core theorems against Mathlib v4.29.1 with no sorry and standard axioms only.